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Non-degeneracy of the harmonic structure on Sierpiński gaskets
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2019 (English)In: Journal of Fractal Geometry, ISSN 2308-1309, Vol. 6, no 2, p. 143-156Article in journal (Refereed) Published
Abstract [en]

We prove that the harmonic extension matrices for the two dimensional level-k Sierpiński gasket are invertible for every k ≥ 2. This has been previously conjectured to be true by Hino in [10] and [11] and tested numerically for k ≤ 50. We also give a necessary condition for the non-degeneracy of the harmonic structure for general finitely ramified self-similar sets based on the vertex connectivity of their first graph approximation.

Place, publisher, year, edition, pages
2019. Vol. 6, no 2, p. 143-156
Keywords [en]
Harmonic structure, harmonic extension matrices, energy Laplacian, Sierpinski gasket, prefractal graphs
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-369738DOI: 10.4171/JFG/73ISI: 000467078300003OAI: oai:DiVA.org:uu-369738DiVA, id: diva2:1271299
Available from: 2018-12-17 Created: 2018-12-17 Last updated: 2019-05-28Bibliographically approved
In thesis
1. Combinatorial and analytical problems for fractals and their graph approximations
Open this publication in new window or tab >>Combinatorial and analytical problems for fractals and their graph approximations
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The recent field of analysis on fractals has been studied under a probabilistic and analytic point of view. In this present work, we will focus on the analytic part developed by Kigami. The fractals we will be studying are finitely ramified self-similar sets, with emphasis on the post-critically finite ones. A prototype of the theory is the Sierpinski gasket. We can approximate the finitely ramified self-similar sets via a sequence of approximating graphs which allows us to use notions from discrete mathematics such as the combinatorial and probabilistic graph Laplacian on finite graphs. Through that approach or via Dirichlet forms, we can define the Laplace operator on the continuous fractal object itself via either a weak definition or as a renormalized limit of the discrete graph Laplacians on the graphs.

The aim of this present work is to study the graphs approximating the fractal and determine connections between the Laplace operator on the discrete graphs and the continuous object, the fractal itself.

In paper I, we study the number of spanning trees on the sequence of graphs approximating a self-similar set admitting spectral decimation.

In paper II, we study harmonic functions on p.c.f. self-similar sets. Unlike the standard Dirichlet problem and harmonic functions in Euclidean space, harmonic functions on these sets may be locally constant without being constant in their entire domain. In that case we say that the fractal has a degenerate harmonic structure. We prove that for a family of variants of the Sierpinski gasket the harmonic structure is non-degenerate.

In paper III, we investigate properties of the Kusuoka measure and the corresponding energy Laplacian on the Sierpinski gaskets of level k.

In papers IV and V, we establish a connection between the discrete combinatorial graph Laplacian determinant and the regularized determinant of the fractal itself. We establish that for a certain class of p.c.f. fractals the logarithm of the regularized determinant appears as a constant in the logarithm of the discrete combinatorial Laplacian.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2019. p. 37
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 112
Keywords
Fractal graphs, energy Laplacian, Kusuoka measure
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-369918 (URN)978-91-506-2739-8 (ISBN)
Public defence
2019-02-15, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2019-01-23 Created: 2018-12-17 Last updated: 2019-01-23

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Tsougkas, Konstantinos

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